Classification of Simple Tracially AF C*-Algebras

We prove that pre-classifiable (see 3.1) simple nuclear tracially AF C*-algebras (TAF) are classified by their K-theory. As a consequence all simple, locally AH and TAF C*-algebras are in fact AH algebras (it is known that there are locally AH algebras that are not AH). We also prove the following Rationalization Theorem. Let A and B be two unital separable nuclear simple TAF C*- algebras with unique normalized traces satisfying the Universal Coefficient Theorem. If A and B have the same (ordered and scaled) K-theory and K0(A)+ is locally finitely generated, then A ⊗ Q ≅ B ⊗ Q, where Q is the UHF-algebra with the rational K0. Classification results (with restriction on K0 - theory) for the above C*-algebras are also obtained. For example, we show that, if A and B are unital nuclear separable simple TAF C*-algebras with the unique normalized trace satisfying the UCT and with K1(A) = K1(B), and A and B have the same rational (scaled ordered) K0, then A ≅ B. Similar results are also obtained for some cases in which K0 is non-divisible such as K0(A) = Z[1/2].